Quantum Hall Ferromagnets-II

Ramin Abolfath Luis Brey Anton Burkov Rene Cote Jim Eisenstein Herb Fertig Steve Girvin Charles Hanna Tomas Jungwirth Kentaro

Nomura Leo Radzihovksy (Doug) Rajaraman Mark Rasolt Jairo Sinova Kun

Yang

Les Houches June 2006

University of Texas at Austin

Skyrmions

Skrymion in 2D O(3) Ferromagnet

QHF Energy Functional

Landau Gauge

Energy Functional

Exchange Interaction

Form Factor

Quantum Hall Skyrmions

Carry Charge !!

Sondhi et al. PRB ‘93

Skyrmions in Ground State- II

Hartree-FockSkyrmion WF

Symmetric GaugeLLL

Wavefunctions

Classical # of reversed spins

Skyrmions Size n=0 LL

Fertig et al. PRB ’94 ‘97

K

g

Fertig et al. PRB ’94Barrett & Tycko PRL ‘95

Skyrmions in Ground State

Skyrmion Crystal

Brey et al. PRL ‘95

Cote et al. PRL ‘97

Skyrmion Crystal Collective Excitations

k1.5 ~ phonons

k1 ~ new

Larmour

QHF Graphene

Nomura cond-mat/06 – Alicea cond-mat/06 – Yang cond-mat/06

Graphene Landau Levels

Conduction

Valence

Semenoff Haldane Kane Mele

Graphene Quantum Hall Effect

Novoselov et al. – Manchester – Nature (2005)

xx

xy

n

Zhang et al. – Columbia – cond-mat/0602649

B= 45 Tesla

SU(4) FerromagnetismArovas/Karlhede (1999) – Ezawa (2002)

A(E)

Landau Band

Exchange Integral

QHF Stoner Criterion

Fogler & Shklovskii (1995); Maude et al. (2005)

AX > 1 QHF

Landau Band Width

InteractionEnergy Scale

Zero Field Conductivity

Vgate

Geim Nature (2005)

Golden Rule

?

Coulomb Scattering I

where

2-3

Coulomb Scattering II

Self-consistent Screening

Das Sarma Ando

Screened Exchange

Phase Diagram

Manchester 1.5Columbia 5.0

Nomura cond-mat/0603 PRL

Broken Symmetry

SU(N)/(SU(M) x SU(N-M)) coset space

2 (N-M) parameters

Graphene Skyrmions - I Graphne Form Factor

Yang et al., cond-mat/0605666

Graphene Skyrmions – II

SU(N)/(SU(M-1) x SU(N-M-1)) coset space

2 (N-M) + 2 (N-1) parameters

Skyrm Lattice has N Goldstone modes

k1.5 & 2 k1 & k2

Next Time:

• Skyrmions

• QHF Anistropy – Ising or XY

• QHF in Graphene

• Quantum Hall Superfluids & Pseudo-spin Transfer Torques